| L(s) = 1 | + 142.·5-s − 217. i·7-s − 25.7i·11-s + 1.39e3·13-s − 4.00e3·17-s − 129. i·19-s − 3.69e3i·23-s + 4.75e3·25-s + 4.08e4·29-s − 4.61e4i·31-s − 3.10e4i·35-s − 6.59e4·37-s + 4.18e4·41-s − 3.00e4i·43-s + 7.97e4i·47-s + ⋯ |
| L(s) = 1 | + 1.14·5-s − 0.634i·7-s − 0.0193i·11-s + 0.632·13-s − 0.814·17-s − 0.0188i·19-s − 0.303i·23-s + 0.304·25-s + 1.67·29-s − 1.54i·31-s − 0.724i·35-s − 1.30·37-s + 0.606·41-s − 0.378i·43-s + 0.768i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.437377094\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.437377094\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 142.T + 1.56e4T^{2} \) |
| 7 | \( 1 + 217. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 25.7iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 1.39e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 4.00e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + 129. iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 3.69e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 4.08e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + 4.61e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 6.59e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 4.18e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + 3.00e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 7.97e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.37e4T + 2.21e10T^{2} \) |
| 59 | \( 1 + 3.00e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 2.43e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 4.06e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 2.49e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 3.66e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 4.52e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 5.12e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 7.97e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + 8.78e3T + 8.32e11T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.581101510736522891973968081536, −8.795608034236306224040058075617, −7.78806084289643143865825205361, −6.62804575546792085550427354284, −6.06773551999369562171360543542, −4.92958626303002100778302843208, −3.92627586509774319630392318409, −2.61446919836158929897019062486, −1.61922368929535729130048815666, −0.47751587717075359808456978065,
1.18119109901314280871945542878, 2.14238423521055680862899990976, 3.12650467177691185098760293352, 4.54874073705140935855183652856, 5.55381050002577806964664063654, 6.25229820833161685101712776211, 7.13937516048860652582595112041, 8.622763668451099716745189645164, 8.917050450139238404483458000568, 10.07054799283904186952710081870
